We will develop and apply a novel univariate neurodegeneration imaging biomarker to brain magnetic resonance images (MRI) obtained from the Alzheimer?s Disease Neuroimaging Initiative (ADNI) dataset and the well-characterized Arizona APOE cohort of presymptomatic individuals. Our recent work has shown that Wasserstein distance-based brain imaging indices outperformed several other univariate brain imaging indices in discriminating Alzheimer?s disease (AD) patients from cognitively unimpaired (CU) subjects with cross- sectional brain MR and fluorodeoxyglucose positron emission tomography (FDG-PET) images. In the current project, we will continue developing novel structural MRI analysis methods based on harmonic maps and the variational principle. Specifically, we will develop 4D harmonic map algorithms to compute canonical imaging spaces of longitudinal brain images and further compute 4D Wasserstein distance-based univariate longitudinal neurodegeneration indices with an efficient variational framework. The proposed system will generate simple, objective, and reliable neurodegeneration imaging biomarkers to quantify progressive presymptomatic anatomical changes related to AD and provide concise and informative univariate outcome measures for randomized clinical trials (RCT). To investigate the reliability and practicality of our method, we will study brain structural MRI scans obtained from the ADNI and the Arizona APOE cohort of presymptomatic subjects. We seek to (1) correlate the computed neurodegeneration imaging indices with longitudinal cognitive trajectories in both ADNI and the independent Arizona APOE cohorts; (2) assess its ability to identify early AD by distinguishing beta-amyloid-positive mild cognitive impairment (MCI)/CU subjects from beta-amyloid-negative MCI/CU subjects in the ADNI cohort; (3) investigate its potential to predict progression rate to the clinical stage of amnestic MCI on CU subjects of the ADNI and the younger presymptomatic individuals of the Arizona APOE cohort; and (4) validate its potential to facilitate the evaluation of AD treatments by reducing the required RCT sample sizes. We will conduct head-to-head comparisons between the proposed univariate neurodegeneration biomarker and other state-of-the-art univariate structural MRI indices with these tasks. We will also develop and freely disseminate our software tools to the research community.